Rethinking Risk: Aspiration as Pure Risk
FUR XII
26 June 2006
Greg B Davies
[email protected]
University College London
INTRODUCTION: THERE IS MORE TO RISK ATTITUDE THAN
DIMINISHING MARGINAL UTILITY
•
Traditional economic theory has had a particularly simple view of risk attitude
–
–
•
No recognition of psychology
–
–
–
•
Not psychologically intuitive
Entirely derived from other effects
Only operational where outcomes are purely numerical
Require restrictive assumptions on EUT
Practical risk measures often ad hoc
–
–
FUR XII
Psychophysics of value
Psychophysics or probability perception
Attitudes to risk itself
Existing theories of risk attitude are inadequate
–
–
–
–
•
Based on Expected Utility Theory
Identified with diminishing marginal utility for wealth
More intuitively sensible
But lack firm theoretical grounding
Copyright © 2006
Page 1
RESULTS FROM EXPERIMENTAL PSYCHOLOGY SUGGEST A
VERY DIFFERENT VALUE FUNCTION
Cumulative Prospect Theory
Value Function
Reference Points
• People evaluate utility as gains or
losses from a reference point not
relative to total wealth
Utility
Loss Aversion
• People are far more sensitive to
losses than to gains
Reference
Point
Gains (£)
Losses (£)
Diminishing Sensitivity
• Weber/Fechner law away from
reference point
• Risk seeking behaviour for losses
Status Quo Bias/Endowment Effect
• People demand more to give up an
object than they are willing to pay
FUR XII
Copyright © 2006
Loss
aversion:
Steeper
for losses
EU = EB[v(x)]
Page 2
IN RANK DEPENDENT UTILITY THEORIES DECISION
WEIGHTS ADD A FURTHER SOURCE OF RISK ATTITUDE
Probability Transformation
Function
1
Underweighting of
probability of middle
outcomes of gamble
Weighting
• Principle of Attention
– Diminishing sensitivity to
probability away from extreme
outcomes
• Psychological interpretation
– Optimism/Hope – Convex
function
– Pessimism/Fear – Concave
Function
“The attention given to an outcome
depends not only on the probability
of the outcomes but also on the
favourability of the outcome in
comparison to the other possible
outcomes” - Diecidue and Wakker
(2001)
FUR XII
Most sensitive
(steepest) at extreme
outcomes: probability
overweighting
0
Copyright © 2006
Cumulative or Decumulative
Probability
1
Page 3
THE BEHAVIOURAL COMPONENTS OF RISK ATTITUDE DO
NOT PROVIDE A COMPLETE CHARACTERISATION
• “Risk Attitude” arises from four
behavioural components
– Utility curvature for gains
– Utility curvature for losses
– Loss aversion
– Distortion of probabilities
Where are attitudes to
“Pure Risk” itself?
“After all the study and all the
clever theorizing, we are left
with a theory of risk taking
that fails to mention risk” Lopes 1987
• None capture intuitive concepts of
“risk”
– First three capture strength of
preference for sure outcomes
– Last captures effects of attention
– All derivative on perceptions of
value and probabilities
• None capture the effect of
introducing risk into a decision
FUR XII
Copyright © 2006
Page 4
PRACTICAL RISK MEASURES REFLECT DISSATISFACTION
WITH STANDARD CONCEPTS OF RISK
• Standard Deviation as a risk measure
– Requires either normal returns distributions or quadratic utility functions
– Not intuitive: positive deviations as weigh as heavily as negative deviations
• Many other measures seem more intuitively plausible
– Semi-standard deviation; VaR; probability of loss; higher order stochastic
dominance; etc...
– But these imply highly restrictive assumptions on utility functions (if they can be
reconciled with EUT at all)
• Reliance on value functions leaves agents with implausible risk attitudes
• We require a way to model attitudes to risk that are not simply the sum of incidental
but unrelated psychophysical responses
FUR XII
Copyright © 2006
Page 5
INTRODUCING PURE RISK ATTITUDES: ALLOWING RISK
ITSELF TO INFLUENCE UTILITY ATTRIBUTION
Example
• Imagine gamble:
(50%, 8 apples; 50%, 0 apples)
• Risk premium is 1 apple: indifference
between gamble and 3 apples for
sure
• If Strength of Preference for gaining 3
apples when I have 0 = that of gaining
5 apples when I have 3
– Diminishing marginal utility
– “Risk attitude” entirely explained
without recourse to “risk”
• Otherwise some aspect of the risk
premium must be due to Risky Utility
FUR XII
• Postulate that attitude to the
introduction of Pure Risk exists
– Primitive concept
– Psychologically intuitive
– Application to all decisions, not
just numerical outcomes
– Distinct from existing concepts
• Traditional measures affect risk
attitude but don’t completely
describe it
• Utilities that represent a “rational”
preference ordering must already
embody all risk attitudes…
Copyright © 2006
Page 6
PURE RISK NEUTRALITY: DECOMPOSITION OF THE
PREFERENCE ORDERING TO ISOLATE PURE RISK ATTITUDE
• Given a role for Pure Risk we can ask what it means to be Pure Risk Neutral
• Postulate a related complete preference ordering which is neutral with regard to
Pure Risk (so excludes any attitudes to Pure Risk)
• PR neutral individual will have a different attribution of utilities uN
• UN to form substrate for Pure Risk theory
• Pure Risk Theory describes the relationship between uN and u
Overall preference
ordering
Pure Risk neutral
preference ordering
Pure Risk preference
ordering
uN
c
NB: Pure Risk by definition does not
include standard risk conceptions
which are included in this portion
FUR XII
Copyright © 2006
u
Portion of allocation due solely
to Pure Risk attitude
Page 7
OUR INTUITIONS OF “RISK” ARE RELATED TO THE
PROBABILITY OF NOT ACHIEVING ASPIRATIONS
max Pr u N   
• Aspirations as Pure Risk
– Minimise probability of bad
outcomes
– Evidence from psychology*,
applied finance**, economics***
– Analogous to VaR in finance
• However this is an incomplete
theory, which relies on an arbitrary
choice of 

* Lopes 1987, Lopes and Oden 1999, Payne 1980,
1981, 2004
Pure Risk Neutral Utility (uN)
** Coombs 1975, Roy 1952, Mao 1970
*** Diecidue & van de Ven 2004, Camerer et al. 1997
FUR XII
Copyright © 2006
Page 8
GENERALISING ASPIRATION POINTS: MULTIPLE POINTS
•
•
•
•
FUR XII
max Pr v  1 
Multiple Aspiration points
– Survival threshold
– Peer group references
– Status quo
Options ranked by cumulative
probability at each point
Lower Aspiration levels of greater
importance?
max Pr v   2 
How to generalise aspiration
notion to arrive at single concept
of Pure Risk?
1
2
Pure Risk Neutral Utility (uN)
Copyright © 2006
Page 9
GENERALISING THE ASPIRATION CONCEPT: ASPIRATION
WEIGHTING FUNCTION
• Take two gambles f and g
– If, for every possible aspiration
point we have:
• Pure Risk can thus be represented
by a non-decreasing function over
uN
Prf u N     Prg u N   
Thus u=(uN)
then we must prefer f to g
• Function governs importance of the
downside uN values in choice
But this is exactly the requirement
of first-order stochastic dominance
applied to uN
• If importance of aspiration levels
decreases with uN,  is concave
so

• If Pure Risk decreases when a
positive constant is added to every
outcome,  is negative exponential

max   u N dF u N   max E  u N 
where  ' u N   0
FUR XII
Copyright © 2006
Page 10
IGNORING PURE RISK ATTITUDES COULD MEAN EXISTING
MODELS AND PREDICTIONS ARE BIASED
• Hitherto value function v(x) used to proxy true EUT utilities u
• But v(x) does not include attitudes to Pure Risk
• Therefore
– Estimates of v(x) parameters mis-specified
– Predictions from outcomes based on v(x) inaccurate
Pure Risk
Neutral
Utilities
Numerical
Outcomes
x
v
uN
Utilities from
Full
Preferences
u
Traditional functions have been trying to force
v(x) to accomplish both psychophysical and
Pure Risk effects
FUR XII
Copyright © 2006
Page 11
PURE RISK ATTITUDE CAN BE USED TO PROVIDE A MORE
COMPLETE ACCOUNT OF AGENTS PREFERENCES
• Instead use v(x) as proxy for uN, not u:
– Thus v(x) is transformed through concave (v(x)): EU = E[(v(x))]
– v(x) includes strength of preference as a source of risk attitude
– Other non-Pure Risk aspects can be incorporated into uN
 Social Preferences
 Regret
 Dynamics (eg, reference point adjustment, house money effect)
 Aspiration
• Decision weighting can be included when arriving at uN
Pure Risk
Neutral
Utilities
Numerical
Outcomes
x
v
Value function (reference point,
curvature, loss aversion)
FUR XII
uN
Missing components of
value function?
Copyright © 2006
Utilities from
Full
Preferences
u
Pure Risk
Attitude
Page 12
PURE RISK INDUCES LOSS AVERSION AND DIFFERENTIAL
CURVATURE WITHOUT ASSUMING EITHER…
Adding Pure Risk Attitudes to Cumulative Prospect Theory Can
Explain Observed Choice Effects With Only Two Parameters
CPT function (1 parameter)
• Equal curvature for gains
and losses
• No loss aversion
3
2
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
-3
-4
-5
Pure Risk and CPT (2 Params)
• Losses more linear than gains
• Displays loss aversion
• Achieving these effects in CPT
requires 4 parameters
-6
CPT Value Function
FUR XII
Copyright © 2006
Pure Risk Utility
Page 13
SUMMARY: PURE RISK ATTITUDE AS AN ADDITIONAL
COMPONENT OF COMPREHENSIVE AGENT PREFERENCES
• Risk attitude now arises from
– Pure Risk attitude (primitive)
– Utility curvature (derived - diminishing sensitivity to monetary amounts)
– Loss Aversion (derived - differential attitudes to gains vs. losses)
– Probability transformation (derived - attention drawn to extreme outcomes)
• Pure Risk attitude derives from a generalisation of Aspiration levels
– Aspiration weighting function transforms Pure Risk neutral utilities
– Function is non-decreasing, concave, negative exponential
• Traditional value function may be used to proxy Pure Risk neutral utilities:
EU = E[(v(x)]
FUR XII
Copyright © 2006
Page 14